Robot arm move a box
To do an animation of this (without using Rigid Body physics) you would simply need to constrain the Box (object being acted upon) to the Clamp of the Robot.
The problem happens when you sit it back down.  You must somehow apply the new transformation but as far as I know, you can't animate Apply (I haven't found a way).

The solution is somewhat fiddly and inelegant and there may be a better way (apart from rigid body), but it works.



1.) Add a Copy Transform Constraint to the Box and select the Clamp as the Object of influence.



2.) For this demo, I set the Clamp's Origin to the location of where I want the Box to be when "clamped".  This is basic so I won't go into detail.  (Ctrl + Shift + Alt + C)

3.) I animated the Clamp to descend onto the box to that point (Shift+S

4.) (gif anim frame 49) At this same frame, keyframe the Constraint Influence to be 0.00 by pressing I.  You could also keyframe the "Eyeball" instead (enable/disable constraint).

5.) (gif anim frame 50) On the very next frame, keyframe an Influence of 1.00. 

6.) (gif anim frame 202) Animate the Clamp however you want and sit the box back down, keyframe the Influence to 1.00 on this frame.  Also, with the box selected, press I and select LocRot.

7.) (gif anim frame 203) On the very next frame, select the box, then Shift Select the Clamp.  Right-Click on both the Rotation and Location Transform menus and choose "Copy All to Selected".  Select the box only in 3D View and press I and select LocRot.  Keyframe the Influence back to 0.00.


This sounds more complicated than it is, I just wanted to get everything in here in case you weren't clear on something.